To hear the shape of a drum is to infer information about the shape of the drumhead from the list of basic harmonics via the use of mathematical theory. The witty title was coined in 1966 by Mark Kac, but these questions can be traced back all the way to Hermann Weyl.

Somewhat more formally, we are given a domain D, typically in the plane but sometimes in higher dimension, and the eigenvalues of a Dirichlet problem for the Laplacian, which we will denote by λ_n. The question is: what can be inferred on D if one knows only the values of λ_n?

It was only in 1992 that Gordon, Webb and Wolpert constructed a pair of regions in the plane, which are not isometric but nevertheless have equal eigenvalues. The regions are non-convex polygons (see picture). The proof that both regions have the same eigenvalues is rather elementary and uses the symmetries of the Laplacian. So, the answer to Kac' question is: for many shapes, one cannot hear the shape of the drum completely. However, some information can be inferred.

Weyl's formula

Weyl's formula states that one can infer the area V of the drum by counting how many of the λ_n-s are quite small. We define N(R) to be the number of eigenvalues smaller than R and we get

where d is the dimension. Weyl also conjectured that the next term in the approximation below would give the perimeter of D. In other words, if A denotes the length of the perimeter (or the surface area in higher dimension), then one should have

where c is some constant that depends only on the dimension. For smooth boundary, this was proved by V. Ja. Ivrii in 1980.

The Weyl-Berry conjecture

For non-smooth boundaries, Berry conjectured in 1979 that the correction should be of the order of R^{D / 2} where D is the Hausdorff dimension of the boundary. This was disproved by J. Brossard and R. A. Carmona who suggested to replace the Hausdorff dimension with the upper box dimension. In the plane, this was proved if the boundary has dimension 1 (1993), but mostly disproved for higher dimensions (1996). Both results are by Lapidus and Pomerance.

References

• Mark Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73:4 (1966), part II, 1-23.
• V. Ja. Ivrii, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary. Funktsional. Anal. i Prilozhen. 14:2 (1980), 25-34 (In Russian).
• Jean Brossard and René Carmona, Can one hear the dimension of a fractal? Comm. Math. Phys. 104:1 (1986), 103-122.
• Michel L. Lapidus, Can one hear the shape of a fractal drum? Partial resolution of the Weyl-Berry conjecture, Geometric analysis and computer graphics (Berkeley, CA, 1988), 119-126, Math. Sci. Res. Inst. Publ., 17, Springer, New York, 1991.
• C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Inventiones mathematicae 110 (1992), 1-22.
• Michel L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture, in: Ordinary and Partial Differential Equations (B. D. Sleeman and R. J. Jarvis, eds.), vol. IV, Proc. Twelfth Internat. Conf. (Dundee, Scotland,UK, June 1992), Pitman Research Notes in Math. Series, vol. 289, Longman and Technical, London, 1993, pp. 126-209.
• Michel L. Lapidus and Carl Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. London Math. Soc. (3) 66:1 (1993), 41-69.
• Michel L. Lapidus and Carl Pomerance, Counterexamples to the modified Weyl-Berry conjecture on fractal drums, Math. Proc. Cambridge Philos. Soc. 119:1 (1996), 167-178.
• M. L. Lapidus and M. van Frankenhuysen, Fractal Geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions, Birkhauser, Boston, 2000. (Revised and enlarged second edition to appear in 2005.)